Combine cranberry juice, sugar, rosemary, and 12 mint leaves in a small saucepan; bring to a boil. Remove from heat; cool to room temperature. Strain juice mixture through a sieve into a bowl; discard solids. Stir in lime juice; cover and refrigerate.
Combine juice mixture, club soda, and rum in a large pitcher; stir well to combine. Serve mixture over crushed ice. Serve immediately.